# Full text of "Treatise On Analysis Vol-Ii"

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```8       XII   TOPOLOGY AND TOPOLOGICAL ALGEBRA

For E - U is closed in E, hence/(E - U) is closed in F by (12.3.6). We have
b £/(E — U) by definition, hence the complement V of/(E — U) in F is an
open neighborhood of b, and clearly U => /~1(V).

PROBLEMS

1.    Find all possible topologies on a set of 2 or 3 elements.

2.    Let % be a set of subsets of a set E. Show that \$ is the set of closed sets for a topology
on E if and only if 3? satisfies the following two conditions: (1) every intersection of a
family of sets belonging to \$ belongs to \$; (2) the empty set belongs to \$, and the union
of two sets belonging to % belongs to %.

3.    Let E be a set and for each x e E let %\$(x) be a set of subsets of E. Then there exists a
topology & on E such that, for all x e E, %\$(x) is the set of neighborhoods of x with
respect to ^", if and only if the sets 85(#) satisfy the following conditions:

(Vj)   Every subset of E which contains a set belonging to %\$(x) belongs to \$(x).

(VH)   The intersection of two sets belonging to %\$(x) belongs to %\$(x).
(Vm)   For all x e E and all V e \$(*), we have x e V.

(VJV)   For all x e E and all V e \$(*), there exists a set W e %(x) such that V e %(y)
for all y e W.

The topology 9" is then unique.

4.    Let A be a commutative ring with an identity element 1 =£ 0. An ideal \$ of A is prime
if A/£ is an integral domain (and therefore ^O). The set of prime ideals of A is called
the spectrum of A and is denoted by Spec(A) (it can be shown to be nonempty). If a is
an ideal in A, the set of prime ideals # => a is denoted by V(a). Show that V(a n 6) =
V(a) u V(fc) for any two ideals a, fc in A. Deduce that the subsets V(a) of Spec(A)
are the closed sets in a topology, called the spectral topology, on Spec(A). If x, y are
two distinct points of Spec(A), show that either there is a neighborhood of x which
does not contain y9 or else there is a neighborhood of y which does not contain x.
Under what conditions is a set {x} consisting of a single point closed in Spec(A)?
Consider the case where A = Z, and the case where A is a discrete valuation ring.

Let A' be another ring and let h: A->A' be a ring homomorphism such that
A(l)— 1. Show that the mapping £'(-» h~\%') of Spec(A') into Spec(A) is continuous
with respect to the spectral topologies.

5.    (a)   The following conditions on a nonempty topological space E are equivalent:
(1) the intersection of any two nonempty open sets in E is nonempty; (2) every non-
empty open set is dense in E; (3) every open set in E is connected. The space E is then
said to be irreducible. A nonempty subset F of a topological space E is said to be an
irreducible set if the subspace F is irreducible.

(b)   Show that in a Hausdorff space every irreducible set consists of a single point.```